In general, the properties of a passive scalar field depend upon both the Reynolds number \(\)  and Schmidt number \(Sc\) , and in many practical problems, both \(\)  and \(Sc\)  are large. For \(Sc>1\) , the smallest scale (in a mean-field sense) in the scalar field is thought to be the Batchelor scale [1]} \(\eta _B=Sc^{-1/2}\eta \) , where \(\eta \)  is the Kolmogorov length scale, and therefore resolving flows with high \(\)  and \(Sc\)  is very challenging using DNS (as well as experiments) due to the spatial and temporal resolution constraints. For problems where the small-scale properties of the scalar field are important, large eddy simulations are not helpful. It is therefore highly desirable to develop models for the small scales of the scalar field that are capable of handling large ranges of \(\)  and \(Sc\) , as well as being computationally efficient. One possibility is to develop Lagrangian models for the scalar gradients in turbulent flows, inspired by the corresponding models for the velocity gradient that have been highly successful both in terms of making predictions and leading to new insights into the small-scale dynamics of turbulent flows [2]}. Analogous models for scalars could be used to explore and understand the small-scale dynamics of scalar fields at \(\)  and \(Sc\)  that are currently far out of reach using either DNS or experiments.
